Complex Variable I
HW #1
MATH 5320 - 1
A. Yu. Solynin
Fall 2012
08.27
Due September 12, 2012
(1) Perform the required calculations and express your answer in the algebraic (cartesian) form
a + ib.
(a) (1(+ i)6 )
(b) ℑ 1+2i
( 3−4i
)
(c) ℜ (1 − i)−2
(d) ℜ
√ ((x + iy)(x − iy))
(e) √i − 1
(f) 4 i
(2) Solve the given equations.
(a) z 3 − 7z 2 + 6z − 10 = 0
(b) z 4 − 2z 2 + 5 = 0
(3) Identify and sketch the set of points satisfying:
(a) |z − (1 + i)| = 1
(b) 2 < |z| < 5
(c) 0 < ℑz < π
(d) |ℜz| + |ℑz| ≤ 1
(e) |z − 2| + |z + 2| = 6
(f) |z − 1| = |z + i|
(g) ℜ(z + 1) = |z − 1|
(h) {w = z 2 : ℜz = 1}
(4) Prove the following:
(a) |ℜz| ≤ |z|
(b) |z + w|2 = |z|2 + |w|2 + 2ℜ(z w̄)
(c) |z + w| ≤ |z| + |w| (Triangle inequality)
(5) Perform the required calculations and express your answer in both algebraic (cartesian) and
polar√forms.
(a) √i − 1
(b) 4 i
(6) Identify and sketch the set of points satisfying:
(a) | arg z| < π/4
(b) {z : |z| < 4, π3 < arg z < 3π
2 }
(c) |z| = arg z
(d) For a fixed b > 0, sketch the curve {eiθ + be−iθ : 0 ≤ θ ≤ 2π}. Hint. Differentiate between
the cases 0 < b < 1, b = 1, and b > 0.
(7) For n > 1, prove that
n+1
(a) 1 + z + z 2 + · · · + z n = 1−z
z ̸= 1. (Does this formula look familiar?)
1−z ,
sin(n+ 1 )θ
2
(b) 1 + cos θ + cos 2θ + · · · + cos nθ = 12 + 2 sin θ/2
(c) The sum of the n-th roots of 1 equals 0.
(8) Prove the following:
(a) If the point P on the sphere corresponds to z under the stereographic projection, then the
antipodal point −P on the sphere corresponds to − z̄1 .
(b) A rotation of the sphere of 180o about the X-axis corresponds under stereographic projection to the inversion z 7→ z1 of C.
(c) ρ(z, ∞) = √ 2 2
1+|z|